Re: opposite of partition

*To*: mathgroup at smc.vnet.net*Subject*: [mg57632] Re: [mg57539] opposite of partition*From*: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>*Date*: Thu, 2 Jun 2005 05:17:34 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

>-----Original Message----- >From: Guy Israeli [mailto:guyi1 at netvision.net.il] To: mathgroup at smc.vnet.net >Sent: Tuesday, May 31, 2005 11:01 AM >Subject: [mg57632] [mg57539] opposite of partition > >Hi, > >How do I do the opposite of partition quickly? > >for example: > >l1= {{a, b, c, d, e}, {f, g, h, i, j}, {k, l, m, n, o}, {p, q, >r, s, t}, {u, >v, w, >x, y}} > >and then if I partition it to blocks will result in > >{{{{a, b}, {f, g}}, {{c, d}, {h, i}}}, {{{k, l}, {p, q}}, {{m, >n}, {r, s}}}} > >flattening it won't help, and its messy to do it by taking all >first lines >of the blocks, then second.. > >How can I do it quickly? > > > >Thanks, > >Guy > > > The opposite of Partition essentially is Join, but we have to understand first what Partition does for nested Lists. This one is easy: In[2]:= l0 = {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z}; In[3]:= l1 = Partition[l0,5] Out[3]= {{a, b, c, d, e}, {f, g, h, i, j}, {k, l, m, n, o}, {p, q, r, s, t}, {u, v, w, x, y}} In[4]:= Apply[Join,l1,{0}] Out[4]= {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y} Clearly you won't get back all of l0 Now for your example: In[5]:= l2=Partition[l1, {2,2}] Out[5]= {{{{a,b},{f,g}},{{c,d},{h,i}}},{{{k,l},{p,q}},{{m,n},{r,s}}}} We have to understand what Partition does here. First it partitions at top level: In[6]:= Map[Partition[#1, 2]&, l1, {0}] Out[6]= {{{a,b,c,d,e},{f,g,h,i,j}},{{k,l,m,n,o},{p,q,r,s,t}}} Then at next level, but which now (due to the first application is not 1 but 1+1 == 2) In[7]:= Map[Partition[#1, 2]&, %, {2}] Out[7]= {{{{a,b},{c,d}},{{f,g},{h,i}}},{{{k,l},{m,n}},{{p,q},{r,s}}}} We can fold both operations to get this: In[8]:= Fold[Map[Function[s,Partition[s, First[#2]]], #1, Rest[#2]]&, l1, Transpose[{{2,2},{0,2}}]] Out[8]= {{{{a,b},{c,d}},{{f,g},{h,i}}},{{{k,l},{m,n}},{{p,q},{r,s}}}} But we have in addition to transpose the levels (in a characteristic fashion): In[9]:= Transpose[%,{1,3,2,4}] Out[9]= {{{{a,b},{f,g}},{{c,d},{h,i}}},{{{k,l},{p,q}},{{m,n},{r,s}}}} In[10]:= %==l2 Out[10]= True So now, as we know the operation, we can reverse each step: In[11]:= l3=Transpose[l2,{1,3,2,4}] Out[11]= {{{{a, b}, {c, d}}, {{f, g}, {h, i}}}, {{{k, l}, {m, n}}, {{p, q}, {r, s}}}} In[12]:= Fold[Apply[Join,##]&,l3,{{2},{0}}] Out[12]= {{a, b, c, d}, {f, g, h, i}, {k, l, m, n}, {p, q, r, s}} This is the best we can restore from original List l1. As this example is a bit special, we make up a more general one: In[20]:= j1=Fold[Partition[#1,#2]&,Range[200],{7,3}] Out[20]= {{{1, 2, 3, 4, 5, 6, 7}, {8, 9, 10, 11, 12, 13, 14}, {15, 16, 17, 18, 19, 20, 21}}, {{22, 23, 24, 25, 26, 27, 28}, {29, 30, 31, 32, 33, 34, 35}, {36, 37, 38, 39, 40, 41, 42}}, {{43, 44, 45, 46, 47, 48, 49}, {50, 51, 52, 53, 54, 55, 56}, {57, 58, 59, 60, 61, 62, 63}}, {{64, 65, 66, 67, 68, 69, 70}, {71, 72, 73, 74, 75, 76, 77}, {78, 79, 80, 81, 82, 83, 84}}, {{85, 86, 87, 88, 89, 90, 91}, {92, 93, 94, 95, 96, 97, 98}, {99, 100, 101, 102, 103, 104, 105}}, {{106, 107, 108, 109, 110, 111, 112}, {113, 114, 115, 116, 117, 118, 119}, {120, 121, 122, 123, 124, 125, 126}}, {{127, 128, 129, 130, 131, 132, 133}, {134, 135, 136, 137, 138, 139, 140}, {141, 142, 143, 144, 145, 146, 147}}, {{148, 149, 150, 151, 152, 153, 154}, {155, 156, 157, 158, 159, 160, 161}, {162, 163, 164, 165, 166, 167, 168}}, {{169, 170, 171, 172, 173, 174, 175}, {176, 177, 178, 179, 180, 181, 182}, {183, 184, 185, 186, 187, 188, 189}}} In[21]:= j2=Partition[j1,{4,2,3}] Out[21]= {{{{{{1, 2, 3}, {8, 9, 10}}, {{22, 23, 24}, {29, 30, 31}}, {{43, 44, 45}, {50, 51, 52}}, {{64, 65, 66}, {71, 72, 73}}}, {{{4, 5, 6}, {11, 12, 13}}, {{25, 26, 27}, {32, 33, 34}}, {{46, 47, 48}, {53, 54, 55}}, {{67, 68, 69}, {74, 75, 76}}}}}, {{{{{85, 86, 87}, {92, 93, 94}}, {{106, 107, 108}, {113, 114, 115}}, {{127, 128, 129}, {134, 135, 136}}, {{148, 149, 150}, {155, 156, 157}}}, {{{88, 89, 90}, {95, 96, 97}}, {{109, 110, 111}, {116, 117, 118}}, {{130, 131, 132}, {137, 138, 139}}, {{151, 152, 153}, {158, 159, 160}}}}}} Again we have to transpose the deeper levels: In[22]:= Transpose[j2,{1,3,5,2,4,6}] Then iterate Join-ing upwards In[23]:= Fold[Apply[Join,##]&,%,{{4},{2},{0}}] Out[23]= {{{1, 2, 3, 4, 5, 6}, {8, 9, 10, 11, 12, 13}}, {{22, 23, 24, 25, 26, 27}, {29, 30, 31, 32, 33, 34}}, {{43, 44, 45, 46, 47, 48}, {50, 51, 52, 53, 54, 55}}, {{64, 65, 66, 67, 68, 69}, {71, 72, 73, 74, 75, 76}}, {{85, 86, 87, 88, 89, 90}, {92, 93, 94, 95, 96, 97}}, {{106, 107, 108, 109, 110, 111}, {113, 114, 115, 116, 117, 118}}, {{127, 128, 129, 130, 131, 132}, {134, 135, 136, 137, 138, 139}}, {{148, 149, 150, 151, 152, 153}, {155, 156, 157, 158, 159, 160}} Again, this is the best we can restore from j1, compare. -- Hartmut Wolf